Not necessarily, i'm not an expert in EU law, but being in the EU they may be forced to recognize the italian ruling, according to the Brussel I Regulation [1].
[1]: https://en.m.wikipedia.org/wiki/Brussels_I_Regulation_2012
[1]: https://en.m.wikipedia.org/wiki/Brussels_I_Regulation_2012
“European Union (EU) law determines which courts of which Member States should hear the case, to avoid conflicting decisions. The general rule is that a person should be sued in the State where s/he is domiciled. Furthermore, other jurisdictional rules may be invoked as alternative in specific cases, for example, the person failing in performance of the contract can be sued at the place of performance of the obligation in question (e.g., in the place where the purchased goods should have been delivered). Special rules exist to protect groups such as consumers, workers and insured persons.”
https://e-justice.europa.eu/85/EN/which_country_s_court_is_r...
So seemingly Italian courts shouldn’t have jurisdiction.